[Math] Proving an entire, complex function with below bounded modulus is constant.

Question

How would I show that an entire function $f$ with the property $|f(z)| \geq 1$ must be constant?

I'm aware of Liouville's theorem, just not sure how to apply it here.

Answer 1

The map $z\mapsto 1/f(z)$ is

• well defined;
• entire;
• bounded on the complex plane.

Here is a generalization.